EXPONENTIAL DECAY OF LEBESGUE NUMBERS
Peng Sun China Economics and Management Academy Central University of Finance and Economics No. 39 College South Road, Beijing, 100081, China
Abstract. We study the exponential rate of decay of Lebesgue numbers of open covers in topological dynamical systems. We show that topological en- tropy is bounded by this rate multiplied by dimension. Some corollaries and examples are discussed.
1. Motivation Entropy, which measures complexity of a dynamical system, has various defini- tions in both topological and measure-theoretical contexts. Most of these definitions are closely related to each other. Given a partition on a measure space, the famous Shannon-McMillan-Breiman Theorem asserts that for almost every point the cell covering it, generated under dynamics, decays in measure with the asymptotic ex- ponential rate equal to the entropy. It is natural to consider analogous objects in topological dynamics. Instead of measurable partitions, the classical definition of topological entropy due to Adler, Konheim and McAndrew involves open covers, which also generate cells under dynamics. We would not like to speak of any in- variant measure as in many cases they may be scarce or pathologic, offering us very little information about the local geometric structure. Diameters of cells are also useless since usually the image of a cell may spread to the whole space. Finally we arrive at Lebesgue number. It measures how large a ball around every point is contained in some cell. It is a global characteristic but exhibits local facts, in which sense catches some idea of Shannon-McMillan-Breiman Theorem. We also notice that the results we obtained provides a good upper estimate of topological entropy which is computable with reasonable effort.
2. Preliminaries on Lebesgue number First we briefly discuss some preliminaries on Lebesgue number and open covers. Some of those can be found in any textbook of elementary topology. For the rest, as well as other facts we discuss in succeeding sections without proof, one can refer to, for example, [5]. The basic result we shall use is the following Lebesgue Covering Lemma. Theorem 2.1. Let (X, d) be a compact metric space. is an open cover of X. Then there is δ > 0 such that every open ball of radiusU at most δ is contained in some element of . We call the largest such number the Lebesgue number of the open cover. U
1991 Mathematics Subject Classification. Primary: 37B40, 54F45. Key words and phrases. Lebesgue number, dimension theory, topological entropy. 1 2 PENG SUN
Proof. If X then the theorem is trivial. If X/ ,∈U let ∈U δ( , x) = sup( inf d(x, y)). U U∈U y∈X\U Then δ( , x) is a continuous function on X taking strictly positive values. Since X is compact,U the function attains its minimum value on X. So δ( ) = min δ( , x) > 0 U x∈X U is the Lebesgue number of the open cover. Remark. Another widely used formulation of Lebesgue Covering Lemma states that there is δ¯ > 0 (the largest one) such that every set of diameter less than δ¯ is contained in some element of . It is easy to see δ δ¯ 2δ. This guarantees that Definition 3.1 is not affected ifU Lebesgue number is≤ defined≤ this way. We have some simple facts on Lebesgue numbers. Lemma 2.2. For two open covers and , we say is finer than , denoted by , if every element of is containedU inV some elementU of . If Vis finer than U≻Vthen δ( ) δ( ). U V U V U ≤ V Lemma 2.3. Let diam( ) = supU∈U diam(U). For every open covers and , if diam( ) <δ( ), then U . U V U V U≻V Lemma 2.4. For any two open covers and , let U V = U V U , V . U V { ∩ | ∈U ∈ V} Then _ δ( ) = min δ( ),δ( ) . U V { U V } Proof. On one hand, is finer_ than and . By Proposition 2.2 U V U V W δ( ) min δ( ),δ( ) U V ≤ { U V } On the other hand, for every x,_ there are U and V such that x ∈U x ∈V δ( , x) = min d(x, y) δ( ), δ( , x) = min d(x, y) δ( ) U y∈X\Ux ≥ U V y∈X\Vx ≥ V Then δ( , x) min d(x, y) = min δ( , x),δ( , x) min δ( ),δ( ) U V ≥ y∈X\(Ux∩Vx) { U V }≥ { U V } _
Now let f be a continuous map from X to itself. Let n−1 n = f −k( ), δ = δ (f, )= δ( n) Uf U n n U Uf k_=0 where f( )= f(U) U . U { | ∈U} Corollary 2.5. Let be an open cover, then U −k δn(f, ) = min δ(f ( )). U 0≤k≤n−1 U Corollary 2.6. If , then for every n, we have f −n( ) f −n( ) and n U ≻V U ≻ V Uf ≻ n, hence δ(f −n( )) δ(f −n( )) and δ (f, ) δ (f, ). Vf U ≤ V n U ≤ n V DECAY OF LEBESGUE NUMBERS 3
3. Decay of Lebesgue numbers Now we turn to the asymptotic decay of Lebesgue numbers. Definition 3.1. Let be an open cover of X. We set U − 1 h (f, ) = lim inf log δn(f, ), L U n→∞ −n U + 1 hL (f, ) = lim sup log δn(f, ), U n→∞ −n U h−(f) = sup h−(f, ) L L U and h+(f) = sup h+(f, ). L L U Here the supremums are taken over all finite open covers. ∗ + − From now on we use hL to denote either hL or hL , when the argument works for both cases. We note that these numbers possess some properties analogous to entropy. Lemma 3.2. For every continuous map f and every open cover , we have: ∗ ∗ U (1) hL(f, ) 0, hence hL(f) 0. U ≥ ∗≥ ∗ (2) If f is an isometry, then hL(f)= hL(f, )=0. − + − + U (3) hL (f, ) hL (f, ), hence hL (f) hL (f). (4) For everyU ≤n > 0, Uh∗ (f, ) = h∗ (f,f≤ −n( )) = h∗ (f, n). If in addition f L U L U L Uf is a homeomorphism, then the first equality also holds for n< 0. (5) If , then h∗ (f, ) h∗ (f, ). U≻V L U ≥ L V ∗ n ∗ Proposition 3.3. For every n> 0, hL(f )= nhL(f). Proof. On one hand, for every finite open cover and m> 0, by Corollary 2.5 U n −jn −j δm(f , ) = min δ(f ( )) min δ(f ( )) = δmn(f, ). U 0≤j≤m−1 U ≥ 0≤j≤mn−1 U U ∗ n ∗ ∗ n ∗ Taking limit we obtain hL(f , ) nhL(f, ), hence hL(f ) nhL(f). On the other hand, U ≤ U ≤ n n −jn n −j δm(f , f ) = min δ(f ( f )) = min δ(f ( )) = δmn(f, ). U 0≤j≤m−1 U 0≤j≤mn−1 U U This implies h∗ (f n, n) nh∗ (f, ). So h∗ (f n) nh∗ (f). L Uf ≥ L U L ≥ L Applying Corollary 2.5, we also have: Corollary 3.4. + 1 −n hL (f, ) lim sup log δ(f ( )). U ≥ n→∞ −n U 1 h−(f, ) lim inf log δ(f −n( )). L U ≥ n→∞ −n U (We intentionally replace f 1−n by f −n in the limits.) In fact, we have: Proposition 3.5.
+ 1 −n hL (f, ) = lim sup log δ(f ( )) U n→∞ −n U − Remark. The analogous result is not necessarily true for hL . 4 PENG SUN
The proposition is a corollary of the following lemma. We also note that Lebesgue number is always bounded by the diameter of the space. Lemma 3.6. Let a K be real numbers, uniformly bounded from below. Let n ≥ b = max a 1 k n . n { k| ≤ ≤ } Then a b lim sup n = lim sup n n→∞ n n→∞ n Proof. By definition, a b . So n ≤ n a b lim sup n lim sup n . n→∞ n ≤ n→∞ n Note that b is a non-decreasing sequence. If there is N such that for all { n} n>N, bn = bN , then b K a lim sup n = 0 = lim lim sup n . n→∞ n n→∞ n ≤ n→∞ n Otherwise, the set J = n b
b bn an a lim sup n = lim sup j = lim sup j lim sup n . n→∞ n j→∞ nj j→∞ nj ≤ n→∞ n
Proposition 3.7. ∗ ∗ hL(f) = lim inf hL(f, ). ǫ→0 diam(U)<ǫ U Proof. By definition, h∗ (f) h∗ (f, ) for every open cover . So L ≥ L U U ∗ ∗ hL(f) lim sup inf hL(f, ). ≥ ǫ→0 diam(U)<ǫ U ∗ ∗ For every θ> 0, if hL(f, ) >hL(f) θ, then for every ǫ<δ( ) and every open cover with diam( ) <ǫ, weV have h∗ (−f, ) h∗ (f, ) >h∗ (f)V θ. So U U L U ≥ L V f − ∗ ∗ hL(f) lim inf inf hL(f, ). ≤ ǫ→0 diam(U)<ǫ U
Remark. This proposition implies that in Definition 3.1 the supremums may be taken over all open covers (not necessarily finite).
Corollary 3.8. For every sequence of open covers k k≥1, if diam( k) 0, then h∗ (f, ) h∗ (f). {U } U → L Uk → L Corollary 3.9. If f is an expansive homeomorphism with expansive constant γ, then for every open cover of diameter less than γ, h∗ (f, )= h∗ (f). U L U L Proof. By assumption, every element of ∞ f n( ) contains at most one point. n=−∞ U is a generator. By [5, Theorem 5.21], for every ǫ > 0 there is N > 0 such that U W diam( N f n( )) < ǫ. But h∗ (f, ) = h∗ (f, N f n( )). So h∗ (f, ) = n=−N U L U L n=−N U L U h∗ (f). L W W DECAY OF LEBESGUE NUMBERS 5
Proposition 3.10. Let X = ∞ f n(X). If x, y X , for each n> 0, let ∞ n=0 ∈ ∞ D(f −n(x),f −n(y)) = inf d(z,z′): f n(z)= x, f n(z′)= y . T { } If X∞ is not a single point, then
− 1 d(x, y) hL (f) sup lim inf log −n −n , ≥ x =y∈X∞ n→∞ n D(f (x),f (y))
+ 1 d(x, y) hL (f) sup lim sup log −n −n . ≥ x =y∈X∞ n→∞ n D(f (x),f (y)) Proof. We only show the first inequality. Proof of the other one is similar. Let 1 d(x, y) sup lim inf log −n −n = λ x,y∈X n→∞ n d(f (x),f (y)) Then for every ǫ> 0, there are x ,y X such that 0 0 ∈ ∞ 1 d(x0,y0) lim inf log −n −n > λ ǫ n→∞ n d(f (x0),f (y0)) −
So there is n0 > 0 such that if n>n0 then
1 d(x0,y0) log −n −n > λ 2ǫ n d(f (x0),f (y0)) −
For any open cover of diameter less than d(x0,y0), y0 is not covered by any U −n −n element of covering x0. So every element of f ( ) covering a point of f (x0) U −n U can not cover any point of f (y0). This implies that δ (f, ) < d(f −n(x ),f −n(y )) n+1 U 0 0 − 1 h (f, ) = lim inf log δn(f, ) λ 2ǫ L U n→∞ −n U ≥ − Apply Proposition 3.7 and let ǫ 0, then we obtain h−(f) λ. → L ≥ Remark. The inequalities may be strict. See Example 6.7.
4. Lebesgue numbers, entropy and dimensions In this section we investigate the relations between decay of Lebesgue numbers, topological entropy and sorts of dimensions. We consider the three definitions of topological entropy: one using open covers, oene using separated sets, and one closely related to Hausdorff dimension. Each of them has something to do with Lebesgue numbers. 4.1. Lebesgue numbers and minimal covers. Denote by S( ) the smallest cardinality of a sub-cover of . For a given continuous map f on aU compact metric space X, the topological entropyU of is U 1 h(f, ) = lim log S( n). U n→∞ n Uf The topological entropy h(f) of f is then defined as the maximum of h(f, ) taking over all finite open covers of X. U Denote by B(x, γ) = y X d(x, y) < γ the open γ-ball around x. Let N(γ) be the minimal number of{ γ∈-balls| needed to} cover X. The upper box dimension of X is defined by log N(γ) dimB(X) = lim sup . γ→0 − log γ 6 PENG SUN
It is clear that if γ γ then N(γ ) N(γ ). 1 ≥ 2 1 ≤ 2 Lemma 4.1. For every open cover , if is a minimal sub-cover then N(δ( )) = S( ). U W W ≥ |W| U Proof. Let = W . Then for each j, there is x W such that x / W for all W { j } j ∈ j j ∈ k k = j. Otherwise Wj is a sub-cover of with smaller cardinality. As every δ( )-ballW−{ is covered} by some elementU of , it can cover at most one W W element in xj 1≤j≤|W|. So the minimal number of δ( )-balls needed to cover X is no less than{ } . W |W| Corollary 4.2. Let ∆( ) = max δ( ) is a minimal sub-cover of . Then N(∆( )) S( ). U { W |W U} U ≥ U Definition 4.3. For every open cover , let ∆ = ∆( n). We set U n Uf ∆ 1 h (f, ) = lim inf log ∆n L U n→∞ −n and ∆ ∆ hL (f) = sup hL (f, ), U U where the supremum is taken over all finite open covers.
∆ − ∆ Remark. For every , we have δ( ) ∆( ). So hL (f, ) hL (f, ) and hL (f) − U U ≥ U U ≥ U ≥ hL (f). Theorem 4.4. dim (X) h∆(f, ) h(f, ). B L U ≥ U Proof. If h(f, ) = 0 then the theorem is trivial since ∆n is non-increasing. U n n If h(f, ) > 0 then as n , S( f ) . But N(∆n) S( f ), we must have ∆ 0. U → ∞ U → ∞ ≥ U n → Fix a small ǫ> 0. For every N0 > 0, by definition of the upper box dimension, there is γ0 > 0 and N1 >N0 such that ∆n <γ0 for all n>N1, and
log N(∆n) < dimB(X)+ ǫ. − log ∆n Hence log ∆ (dim (X)+ ǫ) > log N(∆ ) log S( n), − n B n ≥ Uf 1 1 n lim inf log ∆n dimB(X)+ ǫ lim log S(Uf ). n→∞ −n ≥ n→∞ n Let ǫ 0 then the result follows. → Corollary 4.5. dim (X) h∆(f) h(f). B L ≥ 4.2. Lebesgue numbers and separated sets. The last subsection is just a warm- up. Usually, it is not convenient to refer to Lebesgue number of the minimal cover as it might be quite different from Lebesgue number of the original one. So we shall ∆ − + not focus on hL but turn to hL and hL . For a continuous map f on a compact metric space (X, d), we define for each n a metric n k k df (x, y)= max d(f (x),f (y)). 0≤k≤n−1 DECAY OF LEBESGUE NUMBERS 7
Recall for ǫ > 0, E X is said to be an (n,ǫ)-separated set if dn(x, y) > ǫ for ⊂ f distinct points x, y E. Let sn(f,ǫ) = max E , where the maximum is taken over all (n,ǫ)-separated sets.∈ Then | | 1 1 h(f) = lim lim sup log sn(f,ǫ) = lim lim inf log sn(f,ǫ). ǫ→0 n→∞ n ǫ→0 n→∞ n Now we consider an open cover = U of X. U { α}α∈I Lemma 4.6. For a given open cover and ǫ > 0, if diam( ) < ǫ, then for each n, N(δ (f, )) >s (f,ǫ). U U n U n Proof. Let E be an (n,ǫ)-separated set of cardinality sn(f,ǫ). If distinct points x, y E are covered by the same δ -ball, then the ball is covered by some element ∈ n n−1 V = f −k(U ) n, ik ⊂Uf k\=0 where Uik is some element of for each k. This implies that x, y V and k k ∈ U U k k ∈ f (x),f (y) Uik . Since diam( ) < ǫ, d(f (x),f (y)) < ǫ for 0 k n 1. So n ∈ U ≤ ≤ − df (x, y) <ǫ, which contradicts the fact that x and y are (n,ǫ)-separated. So each δn-ball can cover at most one point in E. N(δn) >sn(f,ǫ). Theorem 4.7. dim (X) h−(f) h(f) B L ≥ Proof. If h(f) = 0 then it is trivial. If h(f) > 0 then for all small ǫ > 0, as n , sn(f,ǫ) . But for every open cover such that diam( ) <ǫ, we have →N(δ ∞) s (f,ǫ→). Hence∞ δ 0. U U n ≥ n n → Fix a small θ > 0. For every N0 > 0, by definition of the upper box dimension, there is γ0 > 0 and N1 >N0 such that δn <γ0 for all n>N1, and
log N(δn) < dimB(X)+ θ − log δn Hence log δ (dim (X)+ θ) > log N(δ ) log s (f,ǫ) − n B n ≥ n 1 1 (dimB(X)+ θ) lim inf log δn(f, ) lim log sn(f,ǫ) n→∞ −n U ≥ n→∞ n This is true for every with diameter less than ǫ and every θ > 0. Let ǫ 0 and apply Proposition 3.7.U → 4.3. Hausdorff dimension and Bowen’s definition of topological entropy. This part has been inspired by [4]. Instead of box dimension we now discuss Haus- dorff dimension. By considering Lebesgue numbers we obtain an inequality relating Hausdorff dimension and topological entropy, which implies [4, Theorem 2.1] and [2, Theorem 2] (See Corollary 5.2). Recall Bowen’s definition of topological entropy [1] that is equivalent to those we have discussed as the space X is assumed to be compact. Let be a finite open cover of X. For a set B X we write B if B is containedU in some element of . Let n (B) be the⊂ largest nonnegative≻ U integer n such that f k(B) for U f,U ≻ U k = 0, 1,...,n 1. If B then n (B) = 0 and if f k(B) for all k then − U f,U ≻ U nf,U (B) = . Now we set diamU (B) = exp( nf,U (B)). If is also a cover of X, we set ∞ − B diamU ( ) = sup diam (B) B B∈B U 8 PENG SUN and for any real number λ, D ( , λ)= (diam (B))λ. U B U BX∈B Then there is a number hU (f) such that
U,λ(X) = lim inf DU ( , λ) is a cover of X and diamU ( ) <ǫ ǫ→0 { B |B B } is for λ
For every ǫ> 0 small enough (much smaller than δn0 (f, )) and every small γ > 0, there is a cover such that diam( ) <ǫ and U B B (diam(B))λ < γ. BX∈B By (2) we have D ( , λK)= (diam (B))λK < (diam(B))λ <γ U B U BX∈B BX∈B while diam ( ) < ǫ1/K. This implies that (X) = 0. Hence λK > h (f), U B U,λK U whenever λ> dim (X) and K>h+( ). So dim (X) h+( ) h (f). H L U H L U ≥ U Remark. Similarly, for Y X we can define (Y ), h (f, Y ), (Y ) and ⊂ U,λ U λ dimH (Y ) (as in [4]). It is not difficult to show that if is a finite open cover of Y , then U dim (Y ) h+(f, ) h (f, Y ). H L U ≥ U Corollary 4.9. dim (X) h+(f) h(f). H L ≥ DECAY OF LEBESGUE NUMBERS 9
5. Lipschitz maps − + We have shown that hL and hL provide upper estimates of topological entropy. Now we show that these numbers are bounded for Lipschitz maps. Recall that a continous map f is Lipschitz with constant L(f) > 0 if for every x, y X, d(f(x),f(y)) L(f) d(x, y). Here we assume that L(f) to be the smallest∈ one among such numbers.≤ Theorem 5.1. If f is Lipschitz with constant L(f), then for every finite open cover , h+(f, ) max log L(f), 0 . U L U ≤ { } Proof. Let L = max L(f), 0 . For every x X and every y B(x, δ( ) L−(n−1)), { } ∈ ∈ U d(f j (x),f j (y)) Lj d(x, y) δ( ) ≤ ≤ U for j =0, 1,...,n 1. This implies that − f j(B(x, δ( ) L−(n−1))) B(f j (x),δ( )) U U ⊂ U ⊂ j for some U , j =0, 1,...,n 1. So δ( n, x) δ( ) L−(n−1) for every x X, j ∈U − Uf ≥ U ∈ hence δ δ( ) L−(n−1). n ≥ U + 1 1 −(n−1) hL (f, ) = lim sup log δn lim sup log(δ( ) L ) = log L. U n→∞ −n ≤ n→∞ −n U
Corollary 5.2. (see also [2][4]) If f is Lipschitz with constant L(f) > 1, then for every Y X, ⊂ dim (Y ) log L(f) h(f, Y ). H ≥ In particular, dim (X) log L(f) h(f). H ≥ Remark. We note (thanks to Anatole Katok) long before Bowen’s definition of topological entropy was introduced, the weaker result involving the box dimension (see, e.g.[3, Theorem 3.2.9]) had been proved by Kushnirenko: dim (X) max log L(f), 0 h(f). B { }≥ Corollary 5.3. If f is Lipschitz, let 1 l(f) = inf log L(f n) n 1 . {n | ≥ } + Then for every finite open cover , hL (f, ) max l(f), 0 . If l(f) > 0, then for every Y UX, U ≤ { } ⊂ dim (Y ) l(f) h(f, Y ). H ≥ n Remark. If f is Lipschitz (L(f) < ), then the sequence log L(f ) 1≤ n≤∞ is sub-additive. In this case ∞ { } 1 l(f) = lim log L(f n). n→∞ n − + In general, hL (f), hL (f) and l(f) may be different from each other. Some examples will be discussed in the next section.
∗ Theorem 5.4. hL(f) is invariant under bi-Lipschitz conjugacy. 10 PENG SUN
Proof. Let H be a bi-Lipschitz conjugacy between f on X and g on Y . For a finite open cover of X and every x X, there is U such that U ∈ ∈U U B(x, δ( )) H−1(B(H(x),δ( ) L(H−1)−1). ⊃ U ⊃ U Then B(H(x),δ( ) L(H−1)−1) H(U) H( ). U ⊂ ∈ U As H is a homeomorphism, this implies (3) δ(H( )) δ( ) L(H−1)−1. U ≥ U Moreover, H is a conjugacy, g−n(H( )) = H(f −n( )). U U Replace by g−n(H( )) in (3), then U U δ(g−n(H( ))) δ(H(f −n( ))) L(H−1)−1 U ≥ U and hence δ (g,H( )) δ (f, ) L(H−1)−1. n U ≥ n U Taking the upper limit we have h∗ (f) h∗ (g). The other direction is the same. L ≥ L ∗ Remark. hL depends on the metric chosen and is not a topological invariant. By the above theorem each of them is the same for strong equivalent metrics ′ (C1d(x, y) < d (x, y) < C2d(x, y)). Box dimension and Hausdorff dimension also depend on the metric. However, the inequalities we obtained holds for any metric, making entropy, a topological invariant, bounded by geometric numbers.
6. Examples To finish this paper, we put here several examples. Example 6.1. Let f : [0, 1] [0, 1] be defined by f(x)= √x. Then h∗ (f)= . → L ∞ Proof. Take any finite open cover with diameter less than 1/10. Then 1/2 / U for every U covering 0. U ∈ ∈U n δ (f, ) δ(f −n( )) f −n(1/2) f −n(0) =2−2 . n U ≤ U ≤ | − | So h∗ (f)= . L ∞ ∗ This example shows that the numbers hL(f) may be unbounded even if h(f) = 0. ∗ −1 ∗ It also shows that hL(f ) may be different from hL(f), when f is a homeomorphism ∗ −1 (here hL(f ) = 0). Example 6.2. Let f : [0, 1] [0, 1] be defined by f(x) = √1 x2, then f is not 2 → ∗ − Lipschitz. But as f (x) = x, we have hL(f, ) = 0 for every open cover . So ∗ U U hL(f) = 0. The following examples show that even for a Lipschitz map f, in the inequality − + hL (f) hL (f) l(f), every relation may be strict. There can be orbits with expanding≤ rates l≤(f) of arbitrary but finite length. So for any fixed open cover the decay will no longer depend on l(f) after finite iteration. These examples illustrate this mechanism and in fact can be modified to be homeomorphisms. DECAY OF LEBESGUE NUMBERS 11
Example 6.3. Let us consider the compact space X = 0 2−m : m =1, 2,... { }∪{ } with the induced metric and topology from R. Define f on X by
k x, x =2−2 for some integer k or x = 0; f(x)= (2x, otherwise. It is easy to check that f is continuous. n+1 n+1 Clearly L(f) = 2. For every large n we have f n(2−2 +1)=2−2 +n+1 so that L(f n) 2n. So l(f) = log 2. On the≥ other hand, denote by X = 0 2−m : m>N . N { }∪{ } If is an open cover of X, then there is an element U0 of and N0 > 0 such that U k0−1 U k0 U0 XN0 . Let k0 be an integer such that 2 N0 < 2 . Then for every n, −n⊃ −n ≤ f (U ) X k0 . As f ( ) is still an open cover, we have 0 ⊃ 2 U k k δ(f −n( )) d(2−2 0 , 2−2 0 +1), U ≥ + which is independent of n. So by Proposition 3.5, hL (f) = 0. Example 6.4. Fix a b> 1. Consider the compact space ≥ −m Xa,b = 0 a : m =1, 2,... { }∪{ p } qa−2 : p N, q N ∪{ a + q ∈ ∈ } p a−2 (1 + bq): p N, q Z, 1+ bq k x, x =2−2 for some integer k or x = 0; f(x)= min y X : y > x , otherwise. ( { ∈ a,b } It is easy to check that f is a homeomorphism. Similar argument as Example ∗ 6.3 shows that l(f) = log a and hL(f) = log b. ∗ This example also shows that hL(f) is far from a topological invariant since all these functions are topologically conjugate for arbitrary values of a and b. Example 6.5. In Example 6.4 we can replace Xa,b by p q X = 0 a−m : m =1, 2,... a−2 ( )±1 : p N, q N . a { }∪{ }∪{ a + q ∈ ∈ } ∗ Then l(f) = log a and hL(f) = 0. This together with Example 6.4 imply that for every l 0, the collection of possible values ≥ h∗ (f): f is a homeomorphism and l(f)= l { L } is the whole interval [0,l]. Moreover, if we consider g = (f,Id) on Xa [0, 1], then we have dimH (X) = ∗ × 1,h(g)=0,l(g) = log a and hL(g) = 0. This shows that Corollary 4.9 can be a strictly better estimate than Corollary 5.3 (also the results in [2][4]). 12 PENG SUN Example 6.6. Let a>b> 0. Consider a sequence sn defined by − { } − 22k 2 22k 1 sn−1 + a, 2 n< 2 for some k N; s1 =0,s2 = a,sn = 2k−1 ≤ 2k ∈ s + b, 22 n< 22 for some k N. ( n−1 ≤ ∈ Then 1 1 lim sup sn = a and lim inf sn = b. n→∞ n n→∞ n Let t = exp( s ), then n − n X = 0, 1 t : n N { }∪{ n ∈ } is a compact metric space with the induced metric from R. Define x, x =0 or x = 1; f(x)= t , x = t ,n N. ( n−1 n ∈ + − Then f is continuous. It is not difficult to see that hL (f)= l(f)= a but hL (f)= b. We can even incorporate the idea of Example 6.4 and obtain examples of homeo- − + morphisms for which the strict inequality hL (f) [1] (MR0338317) R. Bowen Topological entropy for noncompact sets, Trans. Amer. Math. Soc. 184 (1973), 125–136. [2] (MR1667544) X. Dai, Z. Zhou and X. Geng, Some relations between Hausdorff-dimensions and entropies, Sci. China Ser. A 41 (1998), 1068–1075. [3] (MR1326374) A. Katok and B. Hasselblatt, “Introduction to the Modern Theory of Dynamical Systems,” Cambridge University Press, 1995. [4] (MR2018882) M. Misiurewicz, On Bowen’s definition of topological entropy, Discrete Contin. Dyn. Syst. 10 (2004), 827–833. [5] (MR0648108) P. Walters, “An Introduction to Ergodic Theory,” Springer-Verlag, 1982. E-mail address: [email protected]